Answer
$\pi^2$
Work Step by Step
We integrate the integral as follows:
Consider $I=\pi \int_{\pi/4}^{5 \pi/4} (\sin x -\cos x)^2 dx$
or, $=\pi \int_{\pi/4}^{5 \pi/4} (1- 2 \sin x \cos x) dx$
or, $=\pi \int_{\pi/4}^{5 \pi/4} (1- \sin (2x)) dx$
or, $=\pi [x+\dfrac{\cos (2x) }{2}] _{\pi/4}^{5 \pi/4}$
or, $=\pi^2$
